Coordinate transformations

Coordinate Transformations with t

x = space coordinate, t = time coordinate, v = velocity, u = pace

 

Galilean transformation

speed: = xvt and t´ = t,         pace: = xt/u and t´ = t.

Dual Galilean transformation

speed: = tx/v and x´ = x,       pace: = tux and x´ = x.

 

Light:   c := speed of light,        k := pace of light.

speed: x = ct or x/c = t and = ct´, or x´/c = ,

pace:  kx = t or x = t/k and kr′ = or = t´/k.

 

Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with x´ = γ (x − vt) and = γ (txv/c²),

pace:  γ = (1 – k2/u2)–1/2 with γ (xt/u) and t´γ (txk²/u),

which applies only if |v| < |c| or |u| > |k|.

 

Dual Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with λ (t − x/v) and λ (xt (c2/v)),

pace:  λ = (1 – u2/k2)–1/2 with λ (tux) and λ (xt (u/k²)),

which applies only if |v| > |c| or |u| < |k|.

Galileo-Lorentz limit

speed: c1 → ∞: γ1 → 1, t´t,                c2c/2: γ2 → (1 – 4v²/c²)−1/2 and t´γ2 (t – 4x (v/c²)).

pace:  k1 → 0: γ1 → 1, t´t,                 ç2 → 2/k: γ2 → (1 – (1/4) k²/u²)−1/2 and t′ → γ2 (t – (x/4) (k²/u)).

Dual Galileo-Lorentz limit

speed: c1 → 0: λ1 → 1, x´x,               c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and x´λ2 (x – (t/4) (c²/v)).

pace:  k1 → ∞: λ1 → 1, x´x,               k2k/2: λ2 → (1 – 4u²/k²)−1/2 and x´λ2 (x – 4t (u/k²)).

 

If |v| = |c|, then x′ = x and t′ = t.


Coordinate Transformations with τ

x = space coordinate, t = time coordinate, v = velocity, u = pace

c := speed of light,        k := pace of light.

τ := ct                           τ := t/k

 

Galilean transformation

speed: = xτ (v/c) and τ´ = τ,              pace: = xτ (k/u) and τ´ = τ.

Dual Galilean transformation

speed: τ´ = τx (c/v) and x´ = x,              pace: τ´ = τx (u/k) and x´ = x.

 

Light: x = τ and = τ´

 

Lorentz transformation

speed: γ = (1 – v2/c2)–1/2 with = γ (x − τ (v/c)) and τ´ = γ (τx (v/c)),

pace:  γ = (1 – k2/u2)–1/2 with γ (xτ (k/u)) and τ´γ (τx (k/u)),

which applies only if |v| < |c| or |u| > |k|.

 

Dual Lorentz transformation

speed: λ = (1 − c2/v2)–1/2 with τ′λ (τ – x (c/v)) and λ (xτ (c/v)),

pace:  λ = (1 – u2/k2)–1/2 with τ′λ (τx (u/k)) and λ (xτ (u/k)),

which applies only if |v| > |c| or |u| < |k|.

 

Galileo-Lorentz limit

speed: c1 → ∞: γ1 → 1, τ´τ,               c2c/2: γ2 → (1 – 4v²/c²)−1/2 and τ′ → γ2 (τ – 4x (v/c)).

pace:  ç1 → 0: γ1 → 1, τ´τ,                 ç2 → 2/k: γ2 → (1 – (1/4) k²/u²)−1/2 and τ′ → γ2 (τ – (x/4) (k/u)).

Dual Galileo-Lorentz limit

speed: c1 → 0: λ1 → 1, x´x,               c2 → 2/c: λ2 → (1 – (1/4) c²/v²)−1/2 and x´λ2 (x – (τ/4) (c/v)).

pace:  ç1 → ∞: λ1 → 1, x´x,               ç2k/2: λ2 → (1 – 4u²/k²)−1/2 and x´λ2 (x – 4τ (u/k)).

 

If |v| = |c|, then x′ = x and τ′ = τ.


Coordinate Transformations with c := 1

x = space coordinate, t = time coordinate, v = velocity, u = pace

 

Galilean transformation

speed: = xvt and t´ = t,         pace: = xt/u and t´ = t.

Dual Galilean transformation

speed: = tx/v and x´ = x,       pace: = tux and x´ = x.

 

Light:   1 = c := speed of light,  1 = k := pace of light.

speed: x = t and x´ =

pace:  x = t and = .

 

Lorentz transformation

speed: γ = (1 – v2)–1/2 with = γ (x − vt) and = γ (txv),

pace:  γ = (1 – 1/u2)–1/2 with γ (xt/u) and γ (tx/u),

which applies only if |v| < 1 or |u| > 1.

 

Dual Lorentz transformation

speed: λ = (1 − 1/v2)–1/2 with λ (t − x/v) and λ (xt (1/v)),

pace:  λ = (1 – u2)–1/2 with t′λ (tux) and λ (xtu),

which applies only if |v| > 1 or |u| < 1.

 

If |v| = 1, then x´ = x and t´ = t.